3.290 \(\int \frac{1}{(f+g x) (A+B \log (\frac{e (a+b x)^2}{(c+d x)^2}))^2} \, dx\)

Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{1}{(f+g x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2},x\right ) \]

[Out]

Unintegrable[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]

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Rubi [A]  time = 0.071707, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(f+g x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2),x]

[Out]

Defer[Int][1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{(f+g x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx &=\int \frac{1}{(f+g x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.485656, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2),x]

[Out]

Integrate[1/((f + g*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2), x]

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Maple [A]  time = 1.44, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{gx+f} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x+f)/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

[Out]

int(1/(g*x+f)/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b d x^{2} + a c +{\left (b c + a d\right )} x}{2 \,{\left ({\left (b c f - a d f\right )} A B +{\left (b c f \log \left (e\right ) - a d f \log \left (e\right )\right )} B^{2} +{\left ({\left (b c g - a d g\right )} A B +{\left (b c g \log \left (e\right ) - a d g \log \left (e\right )\right )} B^{2}\right )} x + 2 \,{\left ({\left (b c g - a d g\right )} B^{2} x +{\left (b c f - a d f\right )} B^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (b c g - a d g\right )} B^{2} x +{\left (b c f - a d f\right )} B^{2}\right )} \log \left (d x + c\right )\right )}} + \int \frac{b d g x^{2} + 2 \, b d f x + b c f +{\left (d f - c g\right )} a}{2 \,{\left ({\left (b c f^{2} - a d f^{2}\right )} A B +{\left (b c f^{2} \log \left (e\right ) - a d f^{2} \log \left (e\right )\right )} B^{2} +{\left ({\left (b c g^{2} - a d g^{2}\right )} A B +{\left (b c g^{2} \log \left (e\right ) - a d g^{2} \log \left (e\right )\right )} B^{2}\right )} x^{2} + 2 \,{\left ({\left (b c f g - a d f g\right )} A B +{\left (b c f g \log \left (e\right ) - a d f g \log \left (e\right )\right )} B^{2}\right )} x + 2 \,{\left ({\left (b c g^{2} - a d g^{2}\right )} B^{2} x^{2} + 2 \,{\left (b c f g - a d f g\right )} B^{2} x +{\left (b c f^{2} - a d f^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (b c g^{2} - a d g^{2}\right )} B^{2} x^{2} + 2 \,{\left (b c f g - a d f g\right )} B^{2} x +{\left (b c f^{2} - a d f^{2}\right )} B^{2}\right )} \log \left (d x + c\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="maxima")

[Out]

-1/2*(b*d*x^2 + a*c + (b*c + a*d)*x)/((b*c*f - a*d*f)*A*B + (b*c*f*log(e) - a*d*f*log(e))*B^2 + ((b*c*g - a*d*
g)*A*B + (b*c*g*log(e) - a*d*g*log(e))*B^2)*x + 2*((b*c*g - a*d*g)*B^2*x + (b*c*f - a*d*f)*B^2)*log(b*x + a) -
2*((b*c*g - a*d*g)*B^2*x + (b*c*f - a*d*f)*B^2)*log(d*x + c)) + integrate(1/2*(b*d*g*x^2 + 2*b*d*f*x + b*c*f
+ (d*f - c*g)*a)/((b*c*f^2 - a*d*f^2)*A*B + (b*c*f^2*log(e) - a*d*f^2*log(e))*B^2 + ((b*c*g^2 - a*d*g^2)*A*B +
(b*c*g^2*log(e) - a*d*g^2*log(e))*B^2)*x^2 + 2*((b*c*f*g - a*d*f*g)*A*B + (b*c*f*g*log(e) - a*d*f*g*log(e))*B
^2)*x + 2*((b*c*g^2 - a*d*g^2)*B^2*x^2 + 2*(b*c*f*g - a*d*f*g)*B^2*x + (b*c*f^2 - a*d*f^2)*B^2)*log(b*x + a) -
2*((b*c*g^2 - a*d*g^2)*B^2*x^2 + 2*(b*c*f*g - a*d*f*g)*B^2*x + (b*c*f^2 - a*d*f^2)*B^2)*log(d*x + c)), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A^{2} g x + A^{2} f +{\left (B^{2} g x + B^{2} f\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \,{\left (A B g x + A B f\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*g*x + A^2*f + (B^2*g*x + B^2*f)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))
^2 + 2*(A*B*g*x + A*B*f)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)*(B*log((b*x + a)^2*e/(d*x + c)^2) + A)^2), x)